This is not possible in any meaningful way.

In fact the variety you describe defines a smooth cubic threefold $X$ in $\mathbb{P}^4$. By a famous theorem of Clemens and Griffiths these are not even rational varieties over $\mathbb{C}$. This means that these are no way to parametrise the $\mathbb{C}$-points, let alone the $\mathbb{Q}$-points.

Here are some partial attempts which fail.

Firstly this variety $X$ is unirational over $\mathbb{Q}$ which means that it admits a dominant rational map $\mathbb{P}^3 \dashrightarrow X$ over $\mathbb{Q}$. This gives a Zariski dense set of rational points on $X$. But it is impossible to write all rational points this way, even allowing finitely many maps. This follows from the fact that $X(\mathbb{Q})$ is not thin (see Theorem 3.1 of [1]).

Next one expects that $X$ satisfies weak approximation. This means that $X(\mathbb{Q})$ should be dense in $X(\mathbb{R})$ and $X(\mathbb{Q}_p)$ for all primes $p$. So one can ``describe'' rational points by stipulating that you have rational points close to some given collection of real and $p$-adic points. But this is a partial solution, and in any case no one is able to prove that weak approximation holds with current tools.

This is nothing special about your variety $X$. All these properties are expected to hold for any rationally connected variety with a rational point which is not a rational variety. (In general one needs to replace weak approximation by weak weak approximation.)

[1] Demeio, Julian Lawrence. Elliptic fibrations and the Hilbert property. Int. Math. Res. Not. IMRN 2021, no. 13, 10260--10277.